SLOCC Invariants for Multipartite Mixed States Naihuan Jing1,5 ,∗ Ming Li2,3 , Xianqing Li-Jost2 , Tinggui Zhang2 , and Shao-Ming Fei2,4 1

School of Sciences, South China University of Technology, Guangzhou 510640, China 2

3

Department of Mathematics, China University of Petroleum , Qingdao 266555, China

4

arXiv:1404.6852v1 [quant-ph] 28 Apr 2014

5

Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

Abstract We construct a nontrivial set of invariants for any multipartite mixed states under the SLOCC symmetry. These invariants are given by hyperdeterminants and independent from basis change. In particular, a family of d2 invariants for arbitrary d-dimensional even partite mixed states are explicitly given. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w

∗

Electronic address: [email protected]

1

I.

INTRODUCTION

Classification of multipartite states under stochastic local operations and classical communication (SLOCC) has been a central problem in quantum communication and computation. Recently advances have been made for the classification of pure multipartite states under SLOCC [1, 2] and the dimension of the space of homogeneous SLOCC-invariants in a fixed degree is given as a function of the number of qudits. In this work we present a general method to construct polynomial invariants for mixed states under SLOCC. In particular we also derive general invariants under local unitary (LU) symmetry for mixed states. Polynomial invariants have been investigated in [3–6], which allow in principle to determine all the invariants of local unitary transformations. However, in practice it is a daunting task to solve the large system of algebraic equations. Moreover, little is known for mixed multipartite states even for 2 × 2 × 2-systems. In [7] a complete set of 18 polynomial invariants is presented for the locally unitary equivalence of two qubit-mixed states. Partial results have been obtained for three qubits states [8], tripartite pure and mixed states [9], and some generic mixed states [10–12]. Recently the local unitary equivalence problem for multiqubit [13] and general multipartite [14] pure states have also been solved. In general there are limited tools available to resolve the SLOCC symmetry for mixed multipartite states. Ideally it is hoped that a complete set of invariants under local unitary transformations can be found. Nevertheless usually these invariants depend on the detailed expressions of pure state decompositions of a state. For a given state, such pure state decompositions are infinitely many. Particularly when the density matrices are degenerate, the problem becomes more complicated. In this paper, we present a general method to construct non-trivial invariants under SLOCC symmetry group. All invariants found using this method are a priori LU-invariants, as the LU symmetry group is contained in the SLOCC symmetry group. Therefore SLOCCinvariants are generally much fewer than LU-invariants. Very few information is known about the SLOCC-invariants except for that the degree of the invariant subring [1], and even less is known about how to systematically construct the invariants. We will give a general method to construct four families of SLOCC-invariants for any mixed state of even-partite plus an iterative method to construct general fundamental invariants in all dimensions. We show that the Cayley’s two hyperdeterminants [15, 16] can be used to construct such 2

invariants. Cayley’s second hyperdeterminant has been used to study the entanglement measure like concurrence [17–21] and 3-tangles [22]. It also played an important role in classification of multipartite pure states [23–25]. In [26] we proved that the hyperdetermiants give LU-invariants that are independent from the pure state decomposition and base transition. One of our current main results is that the generalized characteristic polynomials give actually SLOCC invariants and we further show that in the even partite cases the Cayley’s first hyperdeterminant also lead to some of the nontrivial SLOCC invariants for mixed states.

II.

HYPER-MATRIX REPRESENTATIONS OF MIXED STATES

Let H be the state space of the mixed multipartite quantum state: H = H1 ⊗H2 ⊗· · ·⊗Hn , (k)

where Hk is a Hilbert space of dimension dk with inner product h , ii . Suppose σi

(i =

0, . . . , d2k −1) are orthonormal basis of hermitian operators in End(Hi ). Here we can take the (i)

Gell-Mann basis, where σ0 = I is the identity. In particular, they are Pauli spin matrices when di = 2. Then ρ can be expressed as ρ=

X

(1)

(n)

ai1 ···in σi1 ⊗ · · · ⊗ σin ,

(1)

i1 ···in

where the n-dimensional matrix or hypermatrix A = (ai1 ···in ) is called the matrix represen(k)

tation of ρ with respect to the bases {σi }. Here the format of the matrix A is d21 × · · · × d2n , and usual rectangular matrices of size m×n are 2-dimensional matrices of format m×n. One can use A = [ρ]σ to denote the matrix representation of ρ with respect to the orthonormal (k)

bases σ = {σi }. Sometimes the reference to σ is omitted if it is clear from the context. We recall the multiplication of the hyper-matrix A of format f1 ×· · ·×fn by a fi ×fi -matrix B as follows: B ∗k A = C, where Ci1 ···in =

fk X

(2)

bik j ai1 ···ik−1 jik+1 ···in .

j=1

This generalizes the notion of matrix multiplication. When A is a regular rectangular matrix, then B ∗1 A = BA, B ∗2 A = AB t and (B ∗1 C∗2 )A = BAC t , where t stands for transpose. 3

It is easy to see that the matrix representation of ρ under basis change from σ to σ ′ can be simply described by [ρ]σ′ = (P1 ∗1 · · · Pn ∗n )[ρ]σ ,

(3) (k)

where the unitary matrix Pk is the transition matrix from the orthonormal basis {σi } to ′(k)

the orthonormal basis {σi } for k = 1, · · · , n. A function F (ρ) is invariant under SLOCC transformation if F (ρ) = F (AρA† ) for any invertible A = A1 ⊗ · · · ⊗ An ∈ G = SL(d1 ) ⊗ · · · ⊗ SL(dn ), the SLOCC symmetry group, where † denote transpose and conjugate. A true SLOCC-invariant should only depend on ρ, i.e., an invariant F (ρ) given in terms of the matrix [ρ] should be independent from the matrix representation [26]. Since our matrix representation is given as an un-normalized form, the SLOCC-invariants can be taken as a relative invariant under the larger group ˆ = GL(d1 ) ⊗ · · · ⊗ GL(dn ), or a projective invariant under G. ˆ G Two hypermatrices of the same format can be added together to give rise to a third hypermatrix by (ai1 ···in ) + (bi1 ···in ) = (ai1 ···in + bi1 ···in ). The scalar product of a hypermatrix by a number is defined as usual. Then the set of hypermatrices of format f1 × · · · × fn forms a vector space of dimension f1 · · · fn . We denote the vector space by Mat(f1 , · · · , fn ). We consider invariants under the SLOCC symmetry. Let us first look at the special example of bipartite case to motivate our later discussion. Suppose A is the (square) matrix representation of the mixed state ρ on H ⊗2 . Using the same method of [26] it is easy to see the following result. Proposition 1 Suppose A is the Bloch matrix representation of the bipartite mixed state ρ on (Cd )⊗2 . Then the coefficients Fi (A), i = 1, 2, ..., d2, of the characteristic polynomials of A: det(λ I − A) = λd − T r(A)λd−1 + · · · + (−1)d det(A) =

d X

λd−i Fi (A)

(4)

i=0

are LU-invariants. In particular, tr(A) and det(A) are LU-invariants of the two-partite state ρ. We remark that when ρ is a bipartite mixed state in arbitrary dimensional space H1 ⊗ H2 , then the Bloch matrix representation A is a rectangular matrix. Then det(AAT ) and 4

det(AT A) are LU-invariants. When m 6= n, it seems that these are the only known SLOCCinvariants according to [1]. P For pure states |φi = i1 ,··· ,in ai1 ···in |i1 · · · in i, we can associate the hypermatrix A(|φi) = (ai1 ···in ), then the format will be simpler. For bipartite pure state |φi, the determinant det(|φi) is clearly a SLOCC-invariant. In the following we extend the result in Proposition 1 to show that Fi (A) are in fact SLOCC-invariants.

III.

HYPERDETERMINANTS AND SLOCC-INVARIANTS

Given any hypermatrix A, one defines the associated multilinear form fA : H1 ⊗· · ·⊗Hn 7→ C given by

X

fA (x(1) , · · · , x(n) ) =

(1)

(n)

ai1 ···in xi1 · · · xin .

(5)

i1 ,··· ,in

The multilinear form f can also be written as a tensor in H1∗ ⊗ · · · ⊗ Hn∗ : fA =

X

(1)

(n)

ai1 ···in fi1 ⊗ · · · ⊗ fin ,

(6)

i1 ,··· ,in (k)

where fi

(k)

(k)

are standard 1-forms on Hk such that fi (ej ) = δij for i, j = 1, · · · , dim(Vi ).

To examine the action of the SLOCC-symmetry on the hypermatrix representation of ρ, we first give the following important lemma. Lemma 2 For fixed A ∈ Mm and C ∈ Mn , let φ : Mm,n −→ Mm,n be the linear map defined by B 7→ ABC. Then det(φ) = det(A)n det(C)m . Proof: Choose a basis B1 , · · · , Bmn in Mm,n . Recall the vector realignment v(B) ≡ vec(B) [27]. For any rectangular matrix B = (bij ) ∈ Mm,n , the column vector realignment is given by v(B) = [b11 , . . . , bm1 , · · · , b1n , . . . , bmn ]t . The vector realignment satisfies the following fundamental property [27]: v(ABC) = (C t ⊗ A)v(B),

(7)

where C t ⊗ A is the Kronecker tensor product. It is also clear that v(B1 ), · · · , v(Bmn ) form a basis of Cmn . Therefore the map φ induces an isomorphic linear operator on Mmn ≃ Cmn by φ(v(B)) = v(ABC) as v( · ) is apparently linear. We will denote the isomorphic map by

5

the same symbol φ. It then follows that φ[v(B1 ), · · · , v(Bmn )] = [v(AB1 C), · · · , v(ABmn C)] = (C t ⊗ A)[v(B1 ), · · · , v(Bmn )], which means that C t ⊗ A is the matrix of the linear operator φ ∈ End(Cmn ). Hence det(φ) = det(C t ⊗ A) = det(C)m det(A)n . Consider a SLOCC symmetry g = A1 ⊗ · · · ⊗ An ∈ G = SL(d1 ) ⊗ · · · ⊗ SL(dn ) on the state space H = H1 ⊗· · ·⊗Hn . This action is given by ρ 7→ gρg † = (A1 ⊗· · ·⊗An )ρ(A†1 ⊗· · ·⊗A†n ). (k)

Let Bk be the matrix of the action of Ak on the orthonormal basis {σi }, 2

(k) Ak σi A†k

=

dk X

(k)

Bkij σi ,

(8)

j=1

then the hypermatrix of gρg † under the action of g ∈ G is seen to be [gρg †] = (B1 ∗1 · · · ∗n−1 Bn ∗n )[ρ].

(9)

One usually writes it as (B1 ⊗ · · · ⊗ Bn )[ρ] when the hypermatrix [ρ] is identified with its associated tensor via (6). This shows that the action of the SLOCC-symmetry on the mixed state ρ is represented by similar hypermatrix multiplications when one introduces the matrix Bk through (8) for each individual action of Ak of the local SLOCC-symmetry g = A1 ⊗ · · · ⊗ An . The great advantage of (9) is the similarity with the basis transition (3). The following result is immediate from Lemma 2 and the action (9). Theorem 3 The action of SLOCC-group SL(d1 ) ⊗ · · · ⊗ SL(dn ) on ρ induces an action of SL(d21 ) × · · · × SL(d2n ) on its hypermatrix representation [ρ]. c = {g = We comment that a similar result also holds for a larger symmetry group SG

A1 ⊗ · · · ⊗ An |det(g) = 1, Ak ∈ GL(dk )} when all di are equal.

Theorem 3 translates the problem of SLOCC-invariants of ρ into a specific problem of invariants of its hypermatrix representation [ρ] under the group SL(d21 ) × · · · × SL(d2n ). This is exactly the classical problem studied long ago by Cayley. Cayley [15] developed the theory of hyperdeterminant as the theory of homogeneous polynomials in matrix elements Ai1 ···in . A hyperdeterminant is a special homogeneous polynomial invariant under the action of SL(n1 ) × · · · × SL(n1 )). Modern account of Cayley’s 6

theory is the beautiful monograph [16]. Cayley studied two types of hyperdeterminants, and each is used in our current work. The Cayley’s (second) hyperdeterminant Det(A) is defined as the resultant of the multilinear form fA , that is, Det(A) is certain polynomial in components of the tensor fA which is zero if and only if the map fA has a non-trivial point where all partial derivatives with respect to the components of its vector arguments vanish (a non-trivial point means that none of the vector arguments are zero). For example, the resultant of a1 x21 + a2 x1 x2 + a2 x22 = 0 is the polynomial a22 − 4a1 a2 . It is known that the hyperdeterminant Det(A) exists for a given format and is unique up to a scalar factor provided that any factor in the format is less than or equal to the sum of the other factors in the format. Cayley’s two hyperdeterminants are all invariant under the group SL(f1 ) ⊗ · · · ⊗ SL(fn ). In fact the hyperdeterminant Det satisfies the following multiplicative property. Suppose A is a hypermatrix of format f1 × · · · × fn and B is any fi × fi matrix, then Det(B ∗i A) = Det(A) det(B)N/fi ,

(10)

where N is the degree of Det(A). Therefore we have the following result. Theorem 4 The hyperdeterminant of the matrix of ρ is a relative invariant under the action of GL(d1 ) ⊗ · · · ⊗ GL(dn ), thus invariant under the SLOCC symmetry group G = SL(d1 ) ⊗ · · · ⊗ SL(dn ). For an even dimensional matrix A = (Ai1 ···i2n ), where 1 ≤ i1 , . . . , i2n ≤ N, the Cayley’s first hyperdeterminant hdet(A) is the following polynomial 1 hdet(A) = N! σ

X

sgn(σ)

1 ,··· ,σ2n ∈SN

where sgn(σ) =

Qn

i=1

N Y

Aσ1 (i),...,σ2n (i) ,

(11)

i=1

sgn(σi ). The hyperdeterminant hdet reduces to the usual determinant

when A is a square matrix. Cayley’s second hyperdeterminant [15] of the format 2 × 2 × 2 for the hypermatrix A with

7

components aijk , i, j, k ∈ {0, 1}, is given by Det(A) = a2000 a2111 + a2001 a2110 + a2010 a2101 + a2100 a2011 −2a000 a001 a110 a111 − 2a000 a010 a101 a111 −2a000 a011 a100 a111 − 2a001 a010 a101 a110 −2a001 a011 a110 a100 − 2a010 a011 a101 a100 +4a000 a011 a101 a110 + 4a001 a010 a100 a111 . This hyperdeterminant can be written in a more compact form by using the Einstein convention and the Levi-Civita symbol εij , with ε00 = ε11 = 0, ε01 = −ε10 = 1; and bkn = (1/2)εilεjm aijk almn , Det(A) = (1/2)εil εjmbij blm . The four-dimensional hyperdeterminant of the format 2 × 2 × 2 × 2 has been given in Ref. [23]. In order to generate more invariants out of the hyperdeterminants, observe that the tensor product In = Id1 ⊗ · · · ⊗ Idn is a special hypermatrix of format d21 × · · · × d2d provided one ˆ one has that views it as a tensor. Moreover for any A = A1 ⊗ · · · ⊗ An ∈ G, (A1 ∗1 · · · ∗n−1 An ∗n )In = A1 ⊗ · · · ⊗ An

(12)

which is also an element of Mat(d21 , · · · , d2n ). Theorem 5 (1) The hyper-characteristic polynomial Det(λIn − ρ) = Det(λIn − [ρ]) of the multipartite mixed state ρ with [ρ] = (Ai1 ···in ) ∈ Mat(d21 , · · · , d2n ), is a SLOCC-invariant polynomial in λ. The coefficients of the characteristic polynomial are all SLOCC-invariants. In particular, the hyper-trace T r(A) and the hyperdeterminant Det(A) of A are two distinguished polynomial SLOCC-invariants. (2) The same results hold for even-partite quantum states when Det is replaced with hdet. Proof: Both hyperdeterminants are proved similarly, so we only consider Det. By previous discussion either the transition matrix or the action of the SLOCC-symmetry is represented by the hyper-matrix multiplication, see (3) and (9): [ρ]σ′ = (P1 ∗1 · · · ∗n−1 ∗Pn ∗n )[ρ]σ , [gρg †] = (B1 ∗1 · · · ∗n−1 ∗Bn ∗n )[ρ].

8

Therefore the hyper-characteristic polynomial of gρg † is Det ((B1 ∗1 · · · ∗n−1 Bn ∗n )(λIn − [ρ])) = det(B1 )N/d1 · · · det(B1 )N/d1 Det(λIn − ρ) = Det(λIn − ρ). This means that f (λ) = Det(λIn − ρ) is a SLOCC-invariant polynomial and every λcoefficients of f (λ) are SLOCC-invariants. Note that the above calculation also proves the independence from base change. We emphasize that in general the state In is not represented by a tensor product of the identity operators except for the special bipartite case. This means that although in formality our result resembles the usual theory of characteristic polynomial, but the result contains substantially more information in general and constitutes a theoretical method to generate non-trivial invariants for the mixed states. As an application of our general method, we give a few examples. Example 1. For two-qubit case, A = (aij ) ∈ M4 . The followings are SLOCC-invariants. T r(A);

det(A);

m12 + m13 + m14 + m23 + m24 + m34 ; m234 + m134 + m124 + m234 , where mij etc are the principal minors, i.e. m12 is the (1, 2)-principal minor of A. N d 2 Example 2. When ρ is a mixed state on H = 2n 1 C , we will get d SLOCC invariants from hdet(λIn − ρ). The four nontrivial coefficients of the following polynomial provides SLOCC-invariants for the mixed 4-qubit state. hdet(A − λI) = 4 Y 1 X sgn(σ) (Aσ1 (i)σ2 (i)σ3 (i)σ4 (i) − λδσ1 (i)σ2 (i) δσ3 (i)σ4 (i) ). 24 3 i=1 σ∈S4

IV.

CONCLUSION AND DISCUSSION

We have introduced a general set of SLOCC-invariants for mixed multipartite states by using hyperdeterminants of the matrix representation in terms of orthonormal bases. We have shown that these invariants are independent of matrix representations. 9

Our method is fundamentally different from previous studies on LU-invariants of pure states, and enjoys favorable property of independence from base change. In fact, our method shows that as long as the transition matrix belongs to SL(n) then the hypermatrix representation of the mixed state ρ can deliver the same result. Even in the case of pure quantum P states, our method is also different. Suppose that a pure state is given by |φi = I aI |Ii, and assume that the transition matrix from |IihJ| to λ-basis is (pIJ,σ ), then the hypermatrix P of ρ = |φihφ| is ( IJ aI a∗J pIJ,σ ), whose hyperdeterminant is quite different from that of (aI ), even in the format. The method of hyperdeterminants can be used for classification of multipartite quantum states besides theoretical classification of SLOCC symmetry. Since the invariants involve with large amount of computation, in practice they should be useful with the help of computer codes. In our opinion one of the useful messages is that many LU-invariants obtained from the hypermatrix representation are actually SLOCC-invariants. Acknowledgments NJ gratefully acknowledges the partial support of Simons Foundation, Humbolt Foundation, NSF, NSFC and MPI for Mathematics in the Sciences in Leipzig during this work.

[1] Gour G and Wallach N R, 2013, Phys. Rev. Lett. 111, 060502; 2010, J. Math Phys. 51, 112201. [2] Yu N, Qiao Y and Sun X, Characterization of multipartite entanglement, arXiv:1401.2627. [3] Albeverio S, Cattaneo L and Persio D L, 2007, Rep. Math. Phys. 60, 167. [4] Fei S M and Jing N, 2005, Phys. Lett. A 342, 77. [5] Grassl M R, R¨ otteler M and Beth T, 1998, Phys. Rev. A. 58, 1833. [6] Rains E M, 2000, IEEE Trans. Inf. Theory. 46, 54. [7] Makhlin Y, 2002, Quant. Info. Proc. 1, 243. [8] Linden N, Popescu S and Sudbery A, 1999, Phys. Rev. Lett. 83, 243. [9] Albeverio S, Cattaneo L, Fei S M and Wang X H, 2005, Int. J. Quant. Inform. 3, 603. [10] Albeverio S, Fei S M and Goswami D, 2005, Phys. Lett. A. 340, 37. [11] Albeverio S, Fei S M, Parashar P and Yang W L, 2003, Phys. Rev. A. 68, 010303. [12] Sun B Z, Fei S M, Li-Jost X Q and Wang Z X, 2006, J. Phys. A. 39, 43.

10

[13] Kraus B, 2010, Phys. Rev. Lett. 104, 020504; 2010, Phys. Rev. A 82, 032121. [14] Liu B, Li J L, Li X and Qiao C F, 2012, Phys. Rev. Lett. 108, 050501. [15] Cayley A, 1845, Cambridge Math. J. 4, 193. [16] Gelfand I M, Kapranov M M and Zelevinsky A V, 1994, Discriminants, Resaultants, and Multidimensional Determinants. Birkhaeuser, Boston. [17] Albeverio S and Fei S M, 2001, J. Opt. B: Quantum Semiclass. Opt. 3, 223. [18] Hill S and Wootters W K, 1997, Phys. Rev. Lett. 78, 5022. [19] Rungta P, Buzek V, Caves C M, Hillery M and Milburn G J, 2001, Phys. Rev. A. 64, 042315. [20] Uhlmann A, 2000, Phys. Rev. A. 62, 032307. [21] Wootters W K, 1998, Phys. Rev. Lett. 80, 2245. [22] Coffman V, Kundu J and Wootters W K, 2000, Phys. Rev. A. 61, 052306. [23] Luque J G and Thibon J Y, 2003, Phys. Rev. A. 67, 042303. [24] Miyake A, 2003, Phys. Rev. A. 67, 012108. [25] Viehmann O, Eltschka C and Siewert J, 2011, Phys. Rev. A 83, 052330. [26] Zhang T G, Jing N, Li-Jost X Q, Zhao M J and Fei S M, 2013, Euro. Phys. J. D 67, 175. [27] Horn R A and Johnson C R, 1994, Topics in matrix analysis. Cambridge University Press, Cambridge.

11

School of Sciences, South China University of Technology, Guangzhou 510640, China 2

3

Department of Mathematics, China University of Petroleum , Qingdao 266555, China

4

arXiv:1404.6852v1 [quant-ph] 28 Apr 2014

5

Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

Abstract We construct a nontrivial set of invariants for any multipartite mixed states under the SLOCC symmetry. These invariants are given by hyperdeterminants and independent from basis change. In particular, a family of d2 invariants for arbitrary d-dimensional even partite mixed states are explicitly given. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w

∗

Electronic address: [email protected]

1

I.

INTRODUCTION

Classification of multipartite states under stochastic local operations and classical communication (SLOCC) has been a central problem in quantum communication and computation. Recently advances have been made for the classification of pure multipartite states under SLOCC [1, 2] and the dimension of the space of homogeneous SLOCC-invariants in a fixed degree is given as a function of the number of qudits. In this work we present a general method to construct polynomial invariants for mixed states under SLOCC. In particular we also derive general invariants under local unitary (LU) symmetry for mixed states. Polynomial invariants have been investigated in [3–6], which allow in principle to determine all the invariants of local unitary transformations. However, in practice it is a daunting task to solve the large system of algebraic equations. Moreover, little is known for mixed multipartite states even for 2 × 2 × 2-systems. In [7] a complete set of 18 polynomial invariants is presented for the locally unitary equivalence of two qubit-mixed states. Partial results have been obtained for three qubits states [8], tripartite pure and mixed states [9], and some generic mixed states [10–12]. Recently the local unitary equivalence problem for multiqubit [13] and general multipartite [14] pure states have also been solved. In general there are limited tools available to resolve the SLOCC symmetry for mixed multipartite states. Ideally it is hoped that a complete set of invariants under local unitary transformations can be found. Nevertheless usually these invariants depend on the detailed expressions of pure state decompositions of a state. For a given state, such pure state decompositions are infinitely many. Particularly when the density matrices are degenerate, the problem becomes more complicated. In this paper, we present a general method to construct non-trivial invariants under SLOCC symmetry group. All invariants found using this method are a priori LU-invariants, as the LU symmetry group is contained in the SLOCC symmetry group. Therefore SLOCCinvariants are generally much fewer than LU-invariants. Very few information is known about the SLOCC-invariants except for that the degree of the invariant subring [1], and even less is known about how to systematically construct the invariants. We will give a general method to construct four families of SLOCC-invariants for any mixed state of even-partite plus an iterative method to construct general fundamental invariants in all dimensions. We show that the Cayley’s two hyperdeterminants [15, 16] can be used to construct such 2

invariants. Cayley’s second hyperdeterminant has been used to study the entanglement measure like concurrence [17–21] and 3-tangles [22]. It also played an important role in classification of multipartite pure states [23–25]. In [26] we proved that the hyperdetermiants give LU-invariants that are independent from the pure state decomposition and base transition. One of our current main results is that the generalized characteristic polynomials give actually SLOCC invariants and we further show that in the even partite cases the Cayley’s first hyperdeterminant also lead to some of the nontrivial SLOCC invariants for mixed states.

II.

HYPER-MATRIX REPRESENTATIONS OF MIXED STATES

Let H be the state space of the mixed multipartite quantum state: H = H1 ⊗H2 ⊗· · ·⊗Hn , (k)

where Hk is a Hilbert space of dimension dk with inner product h , ii . Suppose σi

(i =

0, . . . , d2k −1) are orthonormal basis of hermitian operators in End(Hi ). Here we can take the (i)

Gell-Mann basis, where σ0 = I is the identity. In particular, they are Pauli spin matrices when di = 2. Then ρ can be expressed as ρ=

X

(1)

(n)

ai1 ···in σi1 ⊗ · · · ⊗ σin ,

(1)

i1 ···in

where the n-dimensional matrix or hypermatrix A = (ai1 ···in ) is called the matrix represen(k)

tation of ρ with respect to the bases {σi }. Here the format of the matrix A is d21 × · · · × d2n , and usual rectangular matrices of size m×n are 2-dimensional matrices of format m×n. One can use A = [ρ]σ to denote the matrix representation of ρ with respect to the orthonormal (k)

bases σ = {σi }. Sometimes the reference to σ is omitted if it is clear from the context. We recall the multiplication of the hyper-matrix A of format f1 ×· · ·×fn by a fi ×fi -matrix B as follows: B ∗k A = C, where Ci1 ···in =

fk X

(2)

bik j ai1 ···ik−1 jik+1 ···in .

j=1

This generalizes the notion of matrix multiplication. When A is a regular rectangular matrix, then B ∗1 A = BA, B ∗2 A = AB t and (B ∗1 C∗2 )A = BAC t , where t stands for transpose. 3

It is easy to see that the matrix representation of ρ under basis change from σ to σ ′ can be simply described by [ρ]σ′ = (P1 ∗1 · · · Pn ∗n )[ρ]σ ,

(3) (k)

where the unitary matrix Pk is the transition matrix from the orthonormal basis {σi } to ′(k)

the orthonormal basis {σi } for k = 1, · · · , n. A function F (ρ) is invariant under SLOCC transformation if F (ρ) = F (AρA† ) for any invertible A = A1 ⊗ · · · ⊗ An ∈ G = SL(d1 ) ⊗ · · · ⊗ SL(dn ), the SLOCC symmetry group, where † denote transpose and conjugate. A true SLOCC-invariant should only depend on ρ, i.e., an invariant F (ρ) given in terms of the matrix [ρ] should be independent from the matrix representation [26]. Since our matrix representation is given as an un-normalized form, the SLOCC-invariants can be taken as a relative invariant under the larger group ˆ = GL(d1 ) ⊗ · · · ⊗ GL(dn ), or a projective invariant under G. ˆ G Two hypermatrices of the same format can be added together to give rise to a third hypermatrix by (ai1 ···in ) + (bi1 ···in ) = (ai1 ···in + bi1 ···in ). The scalar product of a hypermatrix by a number is defined as usual. Then the set of hypermatrices of format f1 × · · · × fn forms a vector space of dimension f1 · · · fn . We denote the vector space by Mat(f1 , · · · , fn ). We consider invariants under the SLOCC symmetry. Let us first look at the special example of bipartite case to motivate our later discussion. Suppose A is the (square) matrix representation of the mixed state ρ on H ⊗2 . Using the same method of [26] it is easy to see the following result. Proposition 1 Suppose A is the Bloch matrix representation of the bipartite mixed state ρ on (Cd )⊗2 . Then the coefficients Fi (A), i = 1, 2, ..., d2, of the characteristic polynomials of A: det(λ I − A) = λd − T r(A)λd−1 + · · · + (−1)d det(A) =

d X

λd−i Fi (A)

(4)

i=0

are LU-invariants. In particular, tr(A) and det(A) are LU-invariants of the two-partite state ρ. We remark that when ρ is a bipartite mixed state in arbitrary dimensional space H1 ⊗ H2 , then the Bloch matrix representation A is a rectangular matrix. Then det(AAT ) and 4

det(AT A) are LU-invariants. When m 6= n, it seems that these are the only known SLOCCinvariants according to [1]. P For pure states |φi = i1 ,··· ,in ai1 ···in |i1 · · · in i, we can associate the hypermatrix A(|φi) = (ai1 ···in ), then the format will be simpler. For bipartite pure state |φi, the determinant det(|φi) is clearly a SLOCC-invariant. In the following we extend the result in Proposition 1 to show that Fi (A) are in fact SLOCC-invariants.

III.

HYPERDETERMINANTS AND SLOCC-INVARIANTS

Given any hypermatrix A, one defines the associated multilinear form fA : H1 ⊗· · ·⊗Hn 7→ C given by

X

fA (x(1) , · · · , x(n) ) =

(1)

(n)

ai1 ···in xi1 · · · xin .

(5)

i1 ,··· ,in

The multilinear form f can also be written as a tensor in H1∗ ⊗ · · · ⊗ Hn∗ : fA =

X

(1)

(n)

ai1 ···in fi1 ⊗ · · · ⊗ fin ,

(6)

i1 ,··· ,in (k)

where fi

(k)

(k)

are standard 1-forms on Hk such that fi (ej ) = δij for i, j = 1, · · · , dim(Vi ).

To examine the action of the SLOCC-symmetry on the hypermatrix representation of ρ, we first give the following important lemma. Lemma 2 For fixed A ∈ Mm and C ∈ Mn , let φ : Mm,n −→ Mm,n be the linear map defined by B 7→ ABC. Then det(φ) = det(A)n det(C)m . Proof: Choose a basis B1 , · · · , Bmn in Mm,n . Recall the vector realignment v(B) ≡ vec(B) [27]. For any rectangular matrix B = (bij ) ∈ Mm,n , the column vector realignment is given by v(B) = [b11 , . . . , bm1 , · · · , b1n , . . . , bmn ]t . The vector realignment satisfies the following fundamental property [27]: v(ABC) = (C t ⊗ A)v(B),

(7)

where C t ⊗ A is the Kronecker tensor product. It is also clear that v(B1 ), · · · , v(Bmn ) form a basis of Cmn . Therefore the map φ induces an isomorphic linear operator on Mmn ≃ Cmn by φ(v(B)) = v(ABC) as v( · ) is apparently linear. We will denote the isomorphic map by

5

the same symbol φ. It then follows that φ[v(B1 ), · · · , v(Bmn )] = [v(AB1 C), · · · , v(ABmn C)] = (C t ⊗ A)[v(B1 ), · · · , v(Bmn )], which means that C t ⊗ A is the matrix of the linear operator φ ∈ End(Cmn ). Hence det(φ) = det(C t ⊗ A) = det(C)m det(A)n . Consider a SLOCC symmetry g = A1 ⊗ · · · ⊗ An ∈ G = SL(d1 ) ⊗ · · · ⊗ SL(dn ) on the state space H = H1 ⊗· · ·⊗Hn . This action is given by ρ 7→ gρg † = (A1 ⊗· · ·⊗An )ρ(A†1 ⊗· · ·⊗A†n ). (k)

Let Bk be the matrix of the action of Ak on the orthonormal basis {σi }, 2

(k) Ak σi A†k

=

dk X

(k)

Bkij σi ,

(8)

j=1

then the hypermatrix of gρg † under the action of g ∈ G is seen to be [gρg †] = (B1 ∗1 · · · ∗n−1 Bn ∗n )[ρ].

(9)

One usually writes it as (B1 ⊗ · · · ⊗ Bn )[ρ] when the hypermatrix [ρ] is identified with its associated tensor via (6). This shows that the action of the SLOCC-symmetry on the mixed state ρ is represented by similar hypermatrix multiplications when one introduces the matrix Bk through (8) for each individual action of Ak of the local SLOCC-symmetry g = A1 ⊗ · · · ⊗ An . The great advantage of (9) is the similarity with the basis transition (3). The following result is immediate from Lemma 2 and the action (9). Theorem 3 The action of SLOCC-group SL(d1 ) ⊗ · · · ⊗ SL(dn ) on ρ induces an action of SL(d21 ) × · · · × SL(d2n ) on its hypermatrix representation [ρ]. c = {g = We comment that a similar result also holds for a larger symmetry group SG

A1 ⊗ · · · ⊗ An |det(g) = 1, Ak ∈ GL(dk )} when all di are equal.

Theorem 3 translates the problem of SLOCC-invariants of ρ into a specific problem of invariants of its hypermatrix representation [ρ] under the group SL(d21 ) × · · · × SL(d2n ). This is exactly the classical problem studied long ago by Cayley. Cayley [15] developed the theory of hyperdeterminant as the theory of homogeneous polynomials in matrix elements Ai1 ···in . A hyperdeterminant is a special homogeneous polynomial invariant under the action of SL(n1 ) × · · · × SL(n1 )). Modern account of Cayley’s 6

theory is the beautiful monograph [16]. Cayley studied two types of hyperdeterminants, and each is used in our current work. The Cayley’s (second) hyperdeterminant Det(A) is defined as the resultant of the multilinear form fA , that is, Det(A) is certain polynomial in components of the tensor fA which is zero if and only if the map fA has a non-trivial point where all partial derivatives with respect to the components of its vector arguments vanish (a non-trivial point means that none of the vector arguments are zero). For example, the resultant of a1 x21 + a2 x1 x2 + a2 x22 = 0 is the polynomial a22 − 4a1 a2 . It is known that the hyperdeterminant Det(A) exists for a given format and is unique up to a scalar factor provided that any factor in the format is less than or equal to the sum of the other factors in the format. Cayley’s two hyperdeterminants are all invariant under the group SL(f1 ) ⊗ · · · ⊗ SL(fn ). In fact the hyperdeterminant Det satisfies the following multiplicative property. Suppose A is a hypermatrix of format f1 × · · · × fn and B is any fi × fi matrix, then Det(B ∗i A) = Det(A) det(B)N/fi ,

(10)

where N is the degree of Det(A). Therefore we have the following result. Theorem 4 The hyperdeterminant of the matrix of ρ is a relative invariant under the action of GL(d1 ) ⊗ · · · ⊗ GL(dn ), thus invariant under the SLOCC symmetry group G = SL(d1 ) ⊗ · · · ⊗ SL(dn ). For an even dimensional matrix A = (Ai1 ···i2n ), where 1 ≤ i1 , . . . , i2n ≤ N, the Cayley’s first hyperdeterminant hdet(A) is the following polynomial 1 hdet(A) = N! σ

X

sgn(σ)

1 ,··· ,σ2n ∈SN

where sgn(σ) =

Qn

i=1

N Y

Aσ1 (i),...,σ2n (i) ,

(11)

i=1

sgn(σi ). The hyperdeterminant hdet reduces to the usual determinant

when A is a square matrix. Cayley’s second hyperdeterminant [15] of the format 2 × 2 × 2 for the hypermatrix A with

7

components aijk , i, j, k ∈ {0, 1}, is given by Det(A) = a2000 a2111 + a2001 a2110 + a2010 a2101 + a2100 a2011 −2a000 a001 a110 a111 − 2a000 a010 a101 a111 −2a000 a011 a100 a111 − 2a001 a010 a101 a110 −2a001 a011 a110 a100 − 2a010 a011 a101 a100 +4a000 a011 a101 a110 + 4a001 a010 a100 a111 . This hyperdeterminant can be written in a more compact form by using the Einstein convention and the Levi-Civita symbol εij , with ε00 = ε11 = 0, ε01 = −ε10 = 1; and bkn = (1/2)εilεjm aijk almn , Det(A) = (1/2)εil εjmbij blm . The four-dimensional hyperdeterminant of the format 2 × 2 × 2 × 2 has been given in Ref. [23]. In order to generate more invariants out of the hyperdeterminants, observe that the tensor product In = Id1 ⊗ · · · ⊗ Idn is a special hypermatrix of format d21 × · · · × d2d provided one ˆ one has that views it as a tensor. Moreover for any A = A1 ⊗ · · · ⊗ An ∈ G, (A1 ∗1 · · · ∗n−1 An ∗n )In = A1 ⊗ · · · ⊗ An

(12)

which is also an element of Mat(d21 , · · · , d2n ). Theorem 5 (1) The hyper-characteristic polynomial Det(λIn − ρ) = Det(λIn − [ρ]) of the multipartite mixed state ρ with [ρ] = (Ai1 ···in ) ∈ Mat(d21 , · · · , d2n ), is a SLOCC-invariant polynomial in λ. The coefficients of the characteristic polynomial are all SLOCC-invariants. In particular, the hyper-trace T r(A) and the hyperdeterminant Det(A) of A are two distinguished polynomial SLOCC-invariants. (2) The same results hold for even-partite quantum states when Det is replaced with hdet. Proof: Both hyperdeterminants are proved similarly, so we only consider Det. By previous discussion either the transition matrix or the action of the SLOCC-symmetry is represented by the hyper-matrix multiplication, see (3) and (9): [ρ]σ′ = (P1 ∗1 · · · ∗n−1 ∗Pn ∗n )[ρ]σ , [gρg †] = (B1 ∗1 · · · ∗n−1 ∗Bn ∗n )[ρ].

8

Therefore the hyper-characteristic polynomial of gρg † is Det ((B1 ∗1 · · · ∗n−1 Bn ∗n )(λIn − [ρ])) = det(B1 )N/d1 · · · det(B1 )N/d1 Det(λIn − ρ) = Det(λIn − ρ). This means that f (λ) = Det(λIn − ρ) is a SLOCC-invariant polynomial and every λcoefficients of f (λ) are SLOCC-invariants. Note that the above calculation also proves the independence from base change. We emphasize that in general the state In is not represented by a tensor product of the identity operators except for the special bipartite case. This means that although in formality our result resembles the usual theory of characteristic polynomial, but the result contains substantially more information in general and constitutes a theoretical method to generate non-trivial invariants for the mixed states. As an application of our general method, we give a few examples. Example 1. For two-qubit case, A = (aij ) ∈ M4 . The followings are SLOCC-invariants. T r(A);

det(A);

m12 + m13 + m14 + m23 + m24 + m34 ; m234 + m134 + m124 + m234 , where mij etc are the principal minors, i.e. m12 is the (1, 2)-principal minor of A. N d 2 Example 2. When ρ is a mixed state on H = 2n 1 C , we will get d SLOCC invariants from hdet(λIn − ρ). The four nontrivial coefficients of the following polynomial provides SLOCC-invariants for the mixed 4-qubit state. hdet(A − λI) = 4 Y 1 X sgn(σ) (Aσ1 (i)σ2 (i)σ3 (i)σ4 (i) − λδσ1 (i)σ2 (i) δσ3 (i)σ4 (i) ). 24 3 i=1 σ∈S4

IV.

CONCLUSION AND DISCUSSION

We have introduced a general set of SLOCC-invariants for mixed multipartite states by using hyperdeterminants of the matrix representation in terms of orthonormal bases. We have shown that these invariants are independent of matrix representations. 9

Our method is fundamentally different from previous studies on LU-invariants of pure states, and enjoys favorable property of independence from base change. In fact, our method shows that as long as the transition matrix belongs to SL(n) then the hypermatrix representation of the mixed state ρ can deliver the same result. Even in the case of pure quantum P states, our method is also different. Suppose that a pure state is given by |φi = I aI |Ii, and assume that the transition matrix from |IihJ| to λ-basis is (pIJ,σ ), then the hypermatrix P of ρ = |φihφ| is ( IJ aI a∗J pIJ,σ ), whose hyperdeterminant is quite different from that of (aI ), even in the format. The method of hyperdeterminants can be used for classification of multipartite quantum states besides theoretical classification of SLOCC symmetry. Since the invariants involve with large amount of computation, in practice they should be useful with the help of computer codes. In our opinion one of the useful messages is that many LU-invariants obtained from the hypermatrix representation are actually SLOCC-invariants. Acknowledgments NJ gratefully acknowledges the partial support of Simons Foundation, Humbolt Foundation, NSF, NSFC and MPI for Mathematics in the Sciences in Leipzig during this work.

[1] Gour G and Wallach N R, 2013, Phys. Rev. Lett. 111, 060502; 2010, J. Math Phys. 51, 112201. [2] Yu N, Qiao Y and Sun X, Characterization of multipartite entanglement, arXiv:1401.2627. [3] Albeverio S, Cattaneo L and Persio D L, 2007, Rep. Math. Phys. 60, 167. [4] Fei S M and Jing N, 2005, Phys. Lett. A 342, 77. [5] Grassl M R, R¨ otteler M and Beth T, 1998, Phys. Rev. A. 58, 1833. [6] Rains E M, 2000, IEEE Trans. Inf. Theory. 46, 54. [7] Makhlin Y, 2002, Quant. Info. Proc. 1, 243. [8] Linden N, Popescu S and Sudbery A, 1999, Phys. Rev. Lett. 83, 243. [9] Albeverio S, Cattaneo L, Fei S M and Wang X H, 2005, Int. J. Quant. Inform. 3, 603. [10] Albeverio S, Fei S M and Goswami D, 2005, Phys. Lett. A. 340, 37. [11] Albeverio S, Fei S M, Parashar P and Yang W L, 2003, Phys. Rev. A. 68, 010303. [12] Sun B Z, Fei S M, Li-Jost X Q and Wang Z X, 2006, J. Phys. A. 39, 43.

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[13] Kraus B, 2010, Phys. Rev. Lett. 104, 020504; 2010, Phys. Rev. A 82, 032121. [14] Liu B, Li J L, Li X and Qiao C F, 2012, Phys. Rev. Lett. 108, 050501. [15] Cayley A, 1845, Cambridge Math. J. 4, 193. [16] Gelfand I M, Kapranov M M and Zelevinsky A V, 1994, Discriminants, Resaultants, and Multidimensional Determinants. Birkhaeuser, Boston. [17] Albeverio S and Fei S M, 2001, J. Opt. B: Quantum Semiclass. Opt. 3, 223. [18] Hill S and Wootters W K, 1997, Phys. Rev. Lett. 78, 5022. [19] Rungta P, Buzek V, Caves C M, Hillery M and Milburn G J, 2001, Phys. Rev. A. 64, 042315. [20] Uhlmann A, 2000, Phys. Rev. A. 62, 032307. [21] Wootters W K, 1998, Phys. Rev. Lett. 80, 2245. [22] Coffman V, Kundu J and Wootters W K, 2000, Phys. Rev. A. 61, 052306. [23] Luque J G and Thibon J Y, 2003, Phys. Rev. A. 67, 042303. [24] Miyake A, 2003, Phys. Rev. A. 67, 012108. [25] Viehmann O, Eltschka C and Siewert J, 2011, Phys. Rev. A 83, 052330. [26] Zhang T G, Jing N, Li-Jost X Q, Zhao M J and Fei S M, 2013, Euro. Phys. J. D 67, 175. [27] Horn R A and Johnson C R, 1994, Topics in matrix analysis. Cambridge University Press, Cambridge.

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